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The Random Selection Problem Several mutually distrusting parties wish to select jointly at random an element of a fixed universe. Goal: Protocol such that even if a party cheats, the outcome will not be too “biased”. Applications: Design a protocol where a trusted third-party makes the selection, then replace third-party with random selection protocol.

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Proof Strategy Suppose protocol has ¿ log* n rounds. Show that one of the players can force the output into a “cheating” set of density o(1) with probability 1-o(1). Strategy: induction on game tree…

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3 -> log*n-log*log*n-O(1) To generalize, induct on the game tree… label every node A-WIN, B-WIN, or TIE: WIN – player can violate SC by choosing cheating set randomly. TIE – both players can violate SC with a cheating set of the form R U S, where R is random and S is a small set of non-random elements. The result stops at ~log* n rounds because |S| grows as a tower in the # of rounds.

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Conclusions We provide matching upper and lower bounds (up to a constant factor) for the round complexity of protocols satisfying a natural criterion. Open Problems/Future Work Leverage results for open problems in well-studied multiparty protocols (leader election, collective coin-flipping, and collective sampling). Study the impact of additional constraints required in literature (e.g., simulatability or message length).